What is the integral from 0 to pi/2 of (1+cos(x))^2 ?

The integral from 0 to pi/2 of (1+cos(x))^2 is (3pi)/4+2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to pi/2 of (1+cos(x))^2

\int_{0}^{\frac{π}{2}} (1 + cos (x))^{2} d x

\frac{3 π}{4} + 2

Decimal

4.35619 \dots

Solution steps

\int_{0}^{\frac{π}{2}} (1 + cos (x))^{2} d x

Expand

(1 + cos (x))^{2} : 1 + 2 cos (x) + cos^{2} (x)

= \int_{0}^{\frac{π}{2}} 1 + 2 cos (x) + cos^{2} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{\frac{π}{2}} 1 d x + \int_{0}^{\frac{π}{2}} 2 cos (x) d x + \int_{0}^{\frac{π}{2}} cos^{2} (x) d x

\int_{0}^{\frac{π}{2}} 1 d x = \frac{π}{2}

\int_{0}^{\frac{π}{2}} 2 cos (x) d x = 2

\int_{0}^{\frac{π}{2}} cos^{2} (x) d x = \frac{π}{4}

= \frac{π}{2} + 2 + \frac{π}{4}

Simplify

\frac{π}{2} + 2 + \frac{π}{4} : \frac{3 π}{4} + 2

= \frac{3 π}{4} + 2

\int_{1}^{2} 7 d x

\int_{1}^{4} (x^{2} + 1)^{2} d x

\int_{\frac{1}{2}}^{1} 4^{x} d x

\int_{0}^{1} y - y^{2} d y

\int_{0}^{t} x^{2} e^{- x} d x

What is the integral from 0 to pi/2 of (1+cos(x))^2 ?
The integral from 0 to pi/2 of (1+cos(x))^2 is (3pi)/4+2